I've had quite a few neighbors' birds end up in my yard. If I were you, I'd definitely put up posters in your neighborhood and let your neighbors know. A lovebird flew in my mom's hair one summer, and the owner ended up living like a mile or two away from us--if she's clipped I doubt she'd venture that far, but it's good to put up posters further away too. By the time we got in touch with the owner, the bird had been missing for weeks. I'd post on your local Craigslist too.
Don't give up hope or stop looking! Like others said, keep food and water out, maybe play a looped recording of parakeet chirps if weather doesn't permit the other bird to go outside.
I don't know if it would help, but if your neighbors have outdoor cats, maybe request that they stay indoors for a day or two. One thing that infuriates me to no end is that I have a fenced-in acre of beautiful natural wildlife and all sorts of birds, but all the stupid neighbors' "outdoor" cats terrorize my yard. Can't tell you how many dead birds we've had on our porch, thanks to one particular well intentioned cat. Outdoor cats = lazy owners Imo. But I digress.